Eisenstein's Irreducibility Criterion

Date: 02/26/2003 at 06:35:17
From: Ken Lai
Subject: Eisenstein's Irreducibility Criterion

Is the criterion only a sufficient condition for a polynomial to be
irreducible, since I find that there are many irreducible 
polynomials that do not meet the criterion. Examples are:

a) x^2 + 3x + 4    Here no common prime number exists that divide 
                   coefficients 3 and 4

b) x^2 + 4x + 8    Here p=2 but p^2 divides 8, which violates the 3rd
                   axiom of the criterion.

So it appears to me that that being a sufficient condition, such sets
of irreducible polynomials that meet the E. Criterion are subsets of a 
bigger set of irreducible polynomials. Is my perception correct ?

Thanks and regards,
Ken Lai 

Date: 02/26/2003 at 07:01:27
From: Doctor Jacques
Subject: Re: Eisenstein's Irreducibility Criterion

Hi Ken Lai,

You are correct, the criterion is merely sufficient, not necessary.

Note that, in some cases, you may have an irreducible polynomial that 
does not satisfy the criterion, but you can transform it in such a 
way that the new polynomial does satisify it.

For example, consider your first example:

  x^2 + 3x + 4

If you make the substitution

  x = y + 1

you get

  y^2 + 5x + 5

which does satisfy Eisenstein's criterion. As both polynomials would 
be simultaneously reducible or not (you can also do the substitution 
after factoring), this proves that the initial polynomial is 
irreducible over the rationals.

Of course, you need some luck to use this trick.

Please feel free to write back if you want to discuss this further.

- Doctor Jacques, The Math Forum
  http://mathforum.org/dr.math/